Abstract
A universal method, called pivotal condensation, for calculating stoichiometric factors of chemical reactions is presented. It is based on our new approach for calculating the basis of the kernel of a matrix over the field of rational numbers. This approach is referred to as kernel pivotal condensation (ker pc) and is presented in detail. It has roughly the same complexity as Gaussian elimination, but can be performed without working with fractions. It is also shown how ker pc can be adapted as a tool to solve inhomogeneous linear systems, invert matrices (this is referred to as inv pc) and determine simultaneously the four subspaces (referred to as 4 pc). Besides, the balancing by inspection method, which is widely used in practice to reduce the size of a linear system arising in chemical balancing, is formulated in a mathematical language. When calculating stoichiometric factors of chemical balancing problems with a non-unique solution the natural question arises how to determine a basis that generates all the solutions over the integers. A method, referred to as smitheration, is introduced that permits to determine such an integer basis from a basis over the rational numbers. If there are few solutions this approach is more efficient than calculating a Smith normal form directly. It is convenient to work over principal ideal domains instead of the ring of integers, so that one can treat balancing problems that depend on one parameter as well.
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Notes
This is available, for example, in Mathematica, Maple and Sage. For the complexity of Smith normal form see [21].
This is optional up to the last iteration.
This is optional up to the last iteration.
According to L. Kronecker rational numbers are Menschenwerk and require as such a lot of attention, i.e., brain capacity.
I learnt this from my dad.
The ring \(\mathbb {Z}[n]\) is not a PID and hence Theorem 10.9 cannot be applied.
I now deserve a grade A from Professor Stout (Central College, Pella, IA) for the entire semester!
The Betelgeusian civilization has been saved: all the dangerous \(FG_3\) could be removed from the atmosphere and replaced by the toxic \(E_{6} F\).
To me it looks like the matrix is stretching out its arms and crawling under the fraction bar.
The matrix A is a piece of the Hankel matrix of the tribonacci numbers (entry A000073 in [15]).
This is of course true for ker pc.
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Acknowledgements
I would like thank the undergraduate crowd at the Escola de Química da Universidade Federal do Rio de Janeiro for helping me to think about the matter, mostly by not paying attention to what I said. The paper is an attempt towards a more just and equitable chemistry. Let us bring the wonders of stoichiometry even to the remotest places of the Amazon rain forest without polluting them with consumer electronics. Those who have seen the movie The Great Race know that ‘Push the button, Max!’ is an elitist and dangerous attitude. I would like to thank Benjamin Briggs for pointing out to me the notion of saturatedness. The whole project was triggered by a chat [9] that I had with Ben Grossmann on stackexchange.com. Thanks to Daniel Herden and Chris Seaton for saving me from embarrassments and Ihsen Yengui for a clarification.
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Herbig, HC. Pivotal condensation and chemical balancing. J Math Chem (2024). https://doi.org/10.1007/s10910-024-01594-9
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DOI: https://doi.org/10.1007/s10910-024-01594-9
Keywords
- Chemical balancing
- Homogeneous linear systems
- Pivotal condensation
- Systems of linear Diophantine equations
- Matrix inversion
- Four subspaces